-
Applied Mathematics & Information Sciences 2(1) (2008),
103–121– An International Journalc©2008 Dixie W Publishing
Corporation, U. S. A.
Peristaltic Flow through a Porous Medium in an Annulus:
Application of an Endoscope
Kh.S. Mekheimer1∗ and Y. Abd Elmaboud2
1Mathematics Department, Faculty of Science, Al-Azhar
University, Nasr City,
11884 Cairo, Egypt
E-mail address: kh [email protected] (Kh.S.
Mekheimer)2Mathematics Department, Faculty of Science, Al-Azhar
University (Assuit Branch),
Assuit, Egypt
E-mail address: yass [email protected] (Y. Abd elmaboud)
Received May 15, 2007; Accepted August 11, 2007
Peristaltic transport with long wavelength approximation and low
Reynolds numberassumptions through a porous medium in an annulus
filled with an incompressible vis-cous and Newtonian fluid, is
investigated theoretically. The inner tube is uniform, rigid,while
the outer tube has a sinusoidal wave traveling down its wall. The
flow is inves-tigated in a wave frame of reference moving with
velocity of the wave. The velocitiesand the pressure gradients have
been obtained in terms of the dimensionless flow rateQ, the
amplitude ratio φ, permeability of the porous medium K and the
radius ratio ²(the ratio between the radius of the inner tube and
the radius of the outer). The effectsof porous medium and an
endoscope on the velocities, pressure gradient, pressure riseand
frictional forces on the inner and outer tubes are discussed.
Keywords: Peristaltic flow, endoscope, porous medium, pressure
rise.
1 Introduction
Peristaltic pumping is a form of fluid transport that occurs
when a progressive wave ofarea contraction or expansion propagates
along the length of distensible duct. Peristalsisis an inherent
property of many biological systems having smooth muscle tubes
whichtransports biofluids by its propulsive movements and is found
in the transport of urine fromkidney to the bladder, the movement
of chyme in the gastro-intestinal tract, intra-uterinefluid motion,
vasomotion of the small blood vessels and in many other glandular
ducts.
∗Corresponding author
-
104 Kh.S. Mekheimer and Y. Abd Elmaboud
The mechanism of peristaltic transport has been exploited for
industrial applications likesanitary fluid transport, blood pumps
in heart lung machine and transport of corrosive fluidswhere the
contact of the fluid with the machinery parts is prohibited. A
number of analytical[1-10], numerical and experimental [11-16]
studies of peristaltic flows of different fluidshave been reported.
there are many examples of natural porous media, such as beach
sand,rye bread, wood, filter paper, human lung, etc. A good
biological example of a porousmedium is the pathological situation
of gallstones when they fall down into bile ductsand close them
partially or completely. Physiologically, inflammation of the
gallbladderepithelium often results from low-grade chronic
infection, and this changes the absorptivecharacteristics of the
gallbladder mucous. As a result, cholesterol begins to
precipitate,usually forming many small crystals of cholesterol on
the surface of the inflamed mucousmembrane. These, in turn, act as
nodes for further precipitation of cholesterol, and thecrystals
grow larger. When such crystals fall down the common bile duct they
cause lossof hepatic secretions to the gut, and also cause severe
pain in the gallbladder region as well(Arthur and Guyton [17]).
Flow through a porous medium has been of considerable interest
in recent years, num-ber of workers employing Darcy’s law, Rapits
et al. [18], and Varshney [19] have solvedproblems of the flow of a
viscous fluid through a porous medium bounded by a verticalsurface.
Mekheimer and Al-Arabi [20], studied nonlinear peristaltic
transport of MHDflow through a porous medium and Mekheimer [6]
studied nonlinear peristaltic transportthrough a porous medium in
an inclined planar channel. The aim of the present study isto
investigate fluid mechanics effects of peristaltic transport
through a porous medium ingap between two coaxial tubes, filled
with incompressible Newtonian fluid, the inner tubeis rigid and the
outer one have a wave trains moving independently. A motivation
ofthe present analysis is the hope that such a problem will be
applicable in many clinicalapplications such as the endoscope
problem.
2 Formulation of the Problem
Consider creeping flow of an incompressible Newtonian fluid
through coaxial tubes thegap between them filled with an isotropic
porous medium. The inner tube is rigid and theouter have a
sinusoidal wave traveling down its walls. The geometry of the wall
surface isdescribed in Fig. 2.1, the equations for the radii
are
r′1 = a1, (2.1)
r′2 = a2 + b cos2πλ
(Z ′ − ct′), (2.2)where a1, a2 are the radius of the inner and
the outer tubes, b is the amplitude of the wave,λ is the
wavelength, c is the propagation velocity and t′ is the time.
-
Peristaltic Flow through a Porous Medium in an Annulus 105
]
U
λ
E
D�
D�
Figure 2.1: Geometry of the problem
Introducing a wave frame (r′, z′) moving with velocity c away
from the fixed frame(R′, Z ′) by the transformation
z′ = Z ′ − ct, r′ = R, w′ = W ′ − c, u′ = U ′, (2.3)
where (u′, w′) and (U ′,W ′) are velocity components. After
using these transformationthen the equations of motion are
∂u′
∂r′+
∂w′
∂z′+
u′
r′= 0, (2.4)
ρ
[u′
∂u′
∂r′+ w′
∂u′
∂z′
]= −∂p
′
∂r′+ µ
[∂2u′
∂r′2+
1r′
∂u′
∂r′+
∂2u′
∂z′2− u
′
r′2
]− µ
K ′u′, (2.5)
ρ
[u′
∂w′
∂r′+ w′
∂w′
∂z′
]= −∂p
′
∂z′+ µ
[∂2w′
∂r′2+
1r′
∂w′
∂r′+
∂2w′
∂z′2
]− µ
K ′(w′ + c), (2.6)
where u′ and w′ are the velocity components in the r′ and z′
directions, respectively, ρis the density, p′ is the pressure, µ is
the viscosity, and K ′ permeability of the porousmedium. We
introduce the following nondimensional variables
r =r′
a2, z =
z′
λ, w =
w′
c, u =
λu′
a2c, p =
a22λµc
p′, K =K ′
a22,
r1 =r′1a2
= ², r2 =r′2a2
= 1 + φ cos(2πz),
Re =ρca2µ
, δ =a2λ
, ² =a1a2
,
(2.7)
where φ is the amplitude ratio, Reynolds number Re and δ is the
dimensionles wave num-ber.
To proceed, we non-dimensionalize Eqs. (2.4-2.6), this
yields
1r
∂(ru)∂r
+∂w
∂z= 0, (2.8)
-
106 Kh.S. Mekheimer and Y. Abd Elmaboud
Reδ3[u
∂u
∂r+ w
∂u
∂z
]= −∂p
∂r+ δ2
∂
∂r
(1r
∂(ru)∂r
)+ δ4
∂2u
∂z2− δ
2
Ku, (2.9)
Reδ
[u
∂w
∂r+ w
∂w
∂z
]= −∂p
∂z+
1r
∂
∂r
(r∂w
∂r
)+ δ2
∂2w
∂z2− 1
K(w + 1). (2.10)
Using the long wavelength approximation and dropping terms of
order δ and higher, itfollows from Eqs. (2.8-2.10) that the
appropriate equations describing the flow in the waveframe are
1r
∂(ru)∂r
+∂w
∂z= 0, (2.11)
∂p
∂r= 0, (2.12)
∂p
∂z=
1r
∂
∂r
(r∂w
∂r
)− 1
K(w + 1), (2.13)
Eq.(2.12) shows that p is not a function of r. The corresponding
dimensionless boundaryconditions are
w = −1 at r = r1,w = −1 at r = r2.
(2.14)
The expressions for the velocity profile of the fluid, obtained
as the solutions ofEqs.(2.13) subject to the boundary conditions
(2.14) are given as
w(r, z) = I0(mr)(
b11pzm2b13
)+ K0(mr)
(b12pzm2b13
)− (m
2 + pz)m2
, (2.15)
where
m =1√K
, b11 = K0(mr1)−K0(mr2), b12 = I0(mr2)− I0(mr1),
b13 = I0(mr2)K0(mr1)− I0(mr1)K0(mr2).where I0, K0 are the
modified Bessel function of the first and second kind respectively
oforder 0. The velocity u(r, z) can be obtained from Eq.(2.11)
after using Eq.(2.15), we get
u(r, z) =1
2m4b213rr2
[2pzr′2
(mr(I1(mr)K0(mr1)+I0(mr1)K1(mr))−1
)(mr2b16−1)
+ r2p′zb13{2 + m(mK0(mr2)[r21I2(mr1)− r2I0(mr1)
]− 2b11rI1(mr)− 2rI0(mr1)K1(mr) + I0(mr2)
(mr2K0(mr1) + 2rK1(mr)
−mr21K2(mr1)))}
], (2.16)
whereb14 = I0(mr2)K2(mr1)− I2(mr1)K0(mr2),b15 = I0(mr1)K2(mr2)−
I2(mr2)K0(mr1),
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Peristaltic Flow through a Porous Medium in an Annulus 107
b16 = I1(mr2)K0(mr1) + I0(mr1)K1(mr2),
and the prime (′) denotes differentiation w.r.t z. The
corresponding stream function (u =−(1/r)∂ψ/∂z and w = (1/r)∂ψ/∂r)
is
ψ(r, z) ={m4b13(r21 − r2) + pz[−2 + m(2rb11I1(mr) +
mK0(mr2)(r2I0(mr1)− r21I2(mr1)) + 2rI0(mr1)K1(mr) +
I0(mr2)(mr21K2(mr1)
− r(mrK0(mr1) + 2K1(mr))))]} × 12m4b13 . (2.17)
The instantaneous volume flow rate Q(z) is given by
Q(z) = 2∫ r2
r1
rw(r, z)dr = (r21 − r22) +pz
m4b13
[m2(b14r21 + b15r
22)− 4
], (2.18)
From Eq.(2.18) we get
pz =m4b13
[m2(b14r21 + b15r22)− 4]
× (Q(z)− (r21 − r22)). (2.19)
Following the analysis given by Shapiro et al.[1], the mean
volume flow, Q over aperiod is obtained as
Q = Q(z) + (1 +φ2
2)− ²2, (2.20)
which on using Eq.(2.19) yields
pz =m4b13
[m2(b14r21 + b15r22)− 4]
×(
Q + ²2 − (1 + φ2
2)− (r21 − r22)
). (2.21)
The pressure rise ∆p and the friction force (at the wall) on the
outer and inner tubes areF (o) and F (i) respectively, in a tube of
length L, in their non-dimensional forms, are givenby
∆p =∫ 1
0
pzdz, (2.22)
F (o) =∫ 1
0
r22(−pz)dz, (2.23)
F (i) =∫ 1
0
r21(−pz)dz. (2.24)
Substituting from Eq.(2.21) in Eqs.(2.22 -2.24) with r1 = ², r2
= 1 + φ cos(2πz),we get the pressure rise and the friction force
(at the wall) on the outer and inner tubes.
In the absence of the porous medium and the inner tube (i.e.,K →
∞, r1 = 0), thepressure rise and the outer friction force in this
case take the form
∆p =−8
(1− φ2)7/2{
Q
(1 +
32φ2
)+
φ2
4(φ2 − 16)
}, (2.25)
-
108 Kh.S. Mekheimer and Y. Abd Elmaboud
F (o) =8
(1− φ2)3/2{
Q− 1− φ2
2+ (1− φ2)3/2
}. (2.26)
The results obtained in Eqs.(2.25-2.26) are the same as those
obtained by Shapiro etal.[1].
3 Numerical Results and Discussion
In order to have an estimate of the quantitative effects of the
various parameters in-volved in the results of the present analysis
we using the MATHEMATICA program. Thenumerical evaluations of the
analytical results obtained for ∆p, F (o), F (i), for
differentparameters values [20-23]: ² = 0.32 up to 0.44, L = λ =
8.01 cm, and K = 0.05 up to0.5.
In Fig. 3.1 the variation of dpdz versus z is shown for
different values of K and Q byfixing the other parameters ² = 0.32
(endoscope problem) and φ = 0.4, it is notice that themaximum
amplitude of the pressure gradient dpdz decreases as K and Q
increase. The effectof changing the amplitude ratio φ and radius
ratio ² is indicated in Fig. 3.2, we observe thatthere is an
increase in the maximum amplitude of dpdz when increasing φ and
².
� ��� ��� ��� ��� �]
���
�
��
���
���
���
���
���
���
���
���
GSG] . ����
����
4 ���
4 ���
Figure 3.1: The variation of pressure gradient dpdz
with z for different values of K and Q at ² = 0.32,φ = 0.4.
-
Peristaltic Flow through a Porous Medium in an Annulus 109
� ��� ��� ��� ��� �]
�����
�������������������������������������������
GSG] ε ���� ����
φ ���
φ ���
Figure 3.2: The variation of pressure gradient dpdz
with z for different values of ² and φ at K = 0.1,Q = 0.1.
It is evident from Fig. 3.3 that there is linear relation
between pressure rise and flowrate, also an increase in the flow
rate reduces the pressure rise and thus maximum flowrate is
achieved at zero pressure rise and maximum pressure occurs at zero
flow rate. Thepressure rise decreases as K increases but it
increases as ² increase.Fig. 3.4, depicts the variation of ∆p with
Q at ² = 0.32, for different values of amplituderatio φ and K. An
interesting observation here is that the pressure rise increases
withincrease φ. The pumping regions, peristaltic pumping (Q > 0
and ∆p > 0), augmentedpumping (Q > 0 and ∆p < 0) and
retrograde pumping (Q < 0 and ∆p > 0) are alsoshown in Figs.
3.3-3.4 and it is clear that the peristaltic pumping region becomes
wider asthe radius ratio ², amplitude ratio φ increases.
The variation of ∆p with permeability of the porous medium K for
different valuesflow rate Q and radius ratio ² at φ = 0.4 is
presented in Fig. 3.5. It is observed that therelation between ∆p
and K is non linear relation, ∆p decreases with increase K.
Figs. 3.6 and 3.7 describe the results obtained for the inner
friction F (i) versus the flowrate Q and Fig. 3.8 depicts the
variation of inner friction F (i) versus permeability of theporous
medium K.
-
110 Kh.S. Mekheimer and Y. Abd Elmaboud
�� ���� ���� ���� ���� � ��� ��� ��� ��� �4
����
�
���
���
���
���
∆S
. ���� ����
ε ����
ε ����
Figure 3.3: The variation of ∆p with Q for different values of K
and ² at φ = 0.4.
�� ���� ���� ���� ���� � ��� ��� ��� ��� �4
����
�
���
���
���
���
���
���
���
���
∆S
φ ��� ���
. ����
. ����
Figure 3.4: The variation of ∆p with Q for different values of φ
and K at ² = 0.32.
-
Peristaltic Flow through a Porous Medium in an Annulus 111
� ���� ���� ���� ���� ���.
��
��
���
���
���
���
∆S
4 ��� ���
ε ����
ε ����
Figure 3.5: The variation of ∆p with K for different values of Q
and ² at φ = 0.4.
�� ���� ���� ���� ���� � ��� ��� ��� ��� �4
����
���
���
�
��
) �L�
. ���� ����
ε ����
ε ����
Figure 3.6: The variation of F (i) with Q for different values
of K and ² at φ = 0.4.
-
112 Kh.S. Mekheimer and Y. Abd Elmaboud
�� ���� ���� ���� ���� � ��� ��� ��� ��� �4
����
����
����
�
���
) �L�
φ ��� ���
. ����
. ����
Figure 3.7: The variation of F (i) with Q for different values
of K and φ at ² = 0.38.
� ���� ���� ���� ���� ���.
���
���
���
���
���
���
���
�
) �L�
4 ���� ����
ε ����
ε ����
Figure 3.8: The variation of F (i) with K for different values
of Q and ² at φ = 0.4.
-
Peristaltic Flow through a Porous Medium in an Annulus 113
�� ���� ���� ���� ���� � ��� ��� ��� ��� �4
����
����
����
����
����
���
�
��
���
) �R�
. ���� ����
ε ����
ε ����
Figure 3.9: The variation of F (o) with Q for different values
of K and ² at φ = 0.4.
Figs. 3.9 and 3.10 describe the results for outer friction force
F (o) versus the flowrate Q and Fig. 3.11 depicts the variation of
outer friction F (o) versus permeability of theporous medium K.
Furthermore, the effect of important parameters as Q, K, ² and φ
on the inner and outerfriction force have been investigated.
We notice from these figures that the inner and outer friction
force have the oppositebehavior compared to the pressure rise. The
inner friction force behaves similar to theouter friction force for
the same values of the parameters, moreover the outer friction
forceis greater than the inner friction force at the same values of
the parameters.
The effect of the permeability of the porous medium K on the
contour map of thevelocities w(r, z) and u(r, z) are investigated
in Figs. 3.12 and 3.13. The lighter coloredregions have a higher
velocity than the regions shaded darker. Fig. 3.12 shows that
thevelocity w, increases as permeability of the porous medium K
increases also the heightand the width of the bolus decreases. The
behavior of the streamlines near the walls aresame as the walls.
Fig. 3.13 shows that the velocity u. It is notice that a steepening
ofthe edges in the sinusoidal behavior of u. There is an increase
in the size of the bolus aspermeability of the porous medium K
increase.
-
114 Kh.S. Mekheimer and Y. Abd Elmaboud
�� ���� ���� ���� ���� � ��� ��� ��� ��� �4
����
����
����
����
����
����
�
���
) �R�
φ ��� ���
. ���� . ����
Figure 3.10: The variation of F (o) with Q for different values
of φ and K at ² = 0.38.
� ���� ���� ���� ���� ���.
����
����
����
����
���
���
���
���
) �R�
4 ���� ����
ε ����
ε ����
Figure 3.11: The variation of F (o) with K for different values
of Q and ² at φ = 0.4.
-
Peristaltic Flow through a Porous Medium in an Annulus 115
]
%
U
]
$
U
]
&
U
Figure 3.12: The contour plot for the velocity w(r, z) at Q =
−1, ² = 0.32, φ = 0.2 with(A)K = 0.001, (B) K = 0.005, (C) K = 0.5,
r ∈ [², r2(z)] and z ∈ [0, 1].
-
116 Kh.S. Mekheimer and Y. Abd Elmaboud
z
r
A
z
r
B
z
r
C
Figure 3.13: The contour plot for the velocity u(r, z) at Q =
−1, ² = 0.32, φ = 0.2 with(A)K = 0.001, (B) K = 0.005, (C) K = 0.5,
r ∈ [², r2(z)] and z ∈ [0, 1].
-
Peristaltic Flow through a Porous Medium in an Annulus 117
���� ���� � ��� ��� ��� ��� ��� ��� ���
]
���
���
���
���
���
���
�
���
���
���
���
U
Figure 3.14: Graph of the streamlines for Q = 0.5, ² = 0.32, φ =
0.4 and K = 0.05.
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���
���
���
���
���
���
�
���
���
���
���
U
Figure 3.15: Graph of the streamlines for Q = 0.5, ² = 0.32, φ =
0.4 and K = 0.9.
-
118 Kh.S. Mekheimer and Y. Abd Elmaboud
���� ���� � ��� ��� ��� ��� ��� ��� ���
]
���
���
���
���
���
���
���
���
�
���
���
���
���
U
Figure 3.16: Graph of the streamlines for Q = 0.1, ² = 0.32, φ =
0.4 and K = 0.2.
���� ���� � ��� ��� ��� ��� ��� ��� ���
]
���
���
���
���
���
���
�
���
���
���
���
U
Figure 3.17: Graph of the streamlines for Q = 0.1, ² = 0.44, φ =
0.4 and K = 0.2.
-
Peristaltic Flow through a Porous Medium in an Annulus 119
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]
���
���
���
���
���
���
���
�
���
���
U
Figure 3.18: Graph of the streamlines for Q = 0.6, ² = 0.32, φ =
0.2 and K = 0.1.
���� ���� � ��� ��� ��� ��� ��� ��� ���
]
���
���
���
���
���
���
���
�
���
���
���
���
U
Figure 3.19: Graph of the streamlines for Q = 0.6, ² = 0.32, φ =
0.4 and K = 0.1.
-
120 Kh.S. Mekheimer and Y. Abd Elmaboud
4 Streamlines and Fluid Trapping
The phenomenon of trapping, whereby a bolus (defined as a volume
of fluid boundedby a closed streamlines in the wave frame) is
transported at the wave speed. Figs. 3.14and 3.15 illustrate the
streamline graphs for different values of permeability of the
porousmedium K for other given fixed set of parameters. It is
observed that the permeability ofthe porous medium increases the
velocities which lead to the fluid element spin forms bolusthis
indicate that the bolus appears as K increases.
The effects of the radius ratio ² on the trapping are
illustrated in Figs. 3.16-3.17. It isevident that the the
streamlines near the walls are parallel to the walls when radius
ratio ²is small but by increasing the radius ratio the bolus
appearing. The effects of the amplituderatio φ on the trapping are
illustrated in Figs. 3.18-3.19. It is evident that the size
oftrapping bolus increases with increases φ with fixing other
parameters.
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